Integrand size = 31, antiderivative size = 680 \[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x)) \, dx=\frac {2 b d f g x \sqrt {d-c^2 d x^2}}{5 c \sqrt {1-c^2 x^2}}-\frac {5 b c d f^2 x^2 \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}+\frac {b d g^2 x^2 \sqrt {d-c^2 d x^2}}{32 c \sqrt {1-c^2 x^2}}-\frac {4 b c d f g x^3 \sqrt {d-c^2 d x^2}}{15 \sqrt {1-c^2 x^2}}+\frac {b c^3 d f^2 x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}-\frac {7 b c d g^2 x^4 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d f g x^5 \sqrt {d-c^2 d x^2}}{25 \sqrt {1-c^2 x^2}}+\frac {b c^3 d g^2 x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {1-c^2 x^2}}+\frac {3}{8} d f^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {d g^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{16 c^2}+\frac {1}{8} d g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))+\frac {1}{4} d f^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))+\frac {1}{6} d g^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {2 d f g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{5 c^2}+\frac {3 d f^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{16 b c \sqrt {1-c^2 x^2}}+\frac {d g^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{32 b c^3 \sqrt {1-c^2 x^2}} \]
3/8*d*f^2*x*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)-1/16*d*g^2*x*(a+b*arcsi n(c*x))*(-c^2*d*x^2+d)^(1/2)/c^2+1/8*d*g^2*x^3*(a+b*arcsin(c*x))*(-c^2*d*x ^2+d)^(1/2)+1/4*d*f^2*x*(-c^2*x^2+1)*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2 )+1/6*d*g^2*x^3*(-c^2*x^2+1)*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)-2/5*d* f*g*(-c^2*x^2+1)^2*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/c^2+2/5*b*d*f*g* x*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)-5/16*b*c*d*f^2*x^2*(-c^2*d*x^2 +d)^(1/2)/(-c^2*x^2+1)^(1/2)+1/32*b*d*g^2*x^2*(-c^2*d*x^2+d)^(1/2)/c/(-c^2 *x^2+1)^(1/2)-4/15*b*c*d*f*g*x^3*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+1 /16*b*c^3*d*f^2*x^4*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-7/96*b*c*d*g^2 *x^4*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+2/25*b*c^3*d*f*g*x^5*(-c^2*d* x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+1/36*b*c^3*d*g^2*x^6*(-c^2*d*x^2+d)^(1/2)/ (-c^2*x^2+1)^(1/2)+3/16*d*f^2*(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)/b/c /(-c^2*x^2+1)^(1/2)+1/32*d*g^2*(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)/b/ c^3/(-c^2*x^2+1)^(1/2)
Time = 0.36 (sec) , antiderivative size = 332, normalized size of antiderivative = 0.49 \[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x)) \, dx=\frac {d \sqrt {d-c^2 d x^2} \left (225 a^2 \left (6 c^2 f^2+g^2\right )+b^2 c^2 x \left (450 c^2 f^2 x \left (-5+c^2 x^2\right )+192 f g \left (15-10 c^2 x^2+3 c^4 x^4\right )+25 g^2 x \left (9-21 c^2 x^2+8 c^4 x^4\right )\right )-30 a b c \sqrt {1-c^2 x^2} \left (96 f g \left (-1+c^2 x^2\right )^2+30 c^2 f^2 x \left (-5+2 c^2 x^2\right )+5 g^2 x \left (3-14 c^2 x^2+8 c^4 x^4\right )\right )+30 b \left (15 a \left (6 c^2 f^2+g^2\right )-b c \sqrt {1-c^2 x^2} \left (96 f g \left (-1+c^2 x^2\right )^2+30 c^2 f^2 x \left (-5+2 c^2 x^2\right )+5 g^2 x \left (3-14 c^2 x^2+8 c^4 x^4\right )\right )\right ) \arcsin (c x)+225 b^2 \left (6 c^2 f^2+g^2\right ) \arcsin (c x)^2\right )}{7200 b c^3 \sqrt {1-c^2 x^2}} \]
(d*Sqrt[d - c^2*d*x^2]*(225*a^2*(6*c^2*f^2 + g^2) + b^2*c^2*x*(450*c^2*f^2 *x*(-5 + c^2*x^2) + 192*f*g*(15 - 10*c^2*x^2 + 3*c^4*x^4) + 25*g^2*x*(9 - 21*c^2*x^2 + 8*c^4*x^4)) - 30*a*b*c*Sqrt[1 - c^2*x^2]*(96*f*g*(-1 + c^2*x^ 2)^2 + 30*c^2*f^2*x*(-5 + 2*c^2*x^2) + 5*g^2*x*(3 - 14*c^2*x^2 + 8*c^4*x^4 )) + 30*b*(15*a*(6*c^2*f^2 + g^2) - b*c*Sqrt[1 - c^2*x^2]*(96*f*g*(-1 + c^ 2*x^2)^2 + 30*c^2*f^2*x*(-5 + 2*c^2*x^2) + 5*g^2*x*(3 - 14*c^2*x^2 + 8*c^4 *x^4)))*ArcSin[c*x] + 225*b^2*(6*c^2*f^2 + g^2)*ArcSin[c*x]^2))/(7200*b*c^ 3*Sqrt[1 - c^2*x^2])
Time = 0.95 (sec) , antiderivative size = 367, normalized size of antiderivative = 0.54, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {5276, 5262, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \left (d-c^2 d x^2\right )^{3/2} (f+g x)^2 (a+b \arcsin (c x)) \, dx\) |
| \(\Big \downarrow \) 5276 |
| \(\displaystyle \frac {d \sqrt {d-c^2 d x^2} \int (f+g x)^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))dx}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5262 |
| \(\displaystyle \frac {d \sqrt {d-c^2 d x^2} \int \left (\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x)) f^2+2 g x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x)) f+g^2 x^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))\right )dx}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d \sqrt {d-c^2 d x^2} \left (\frac {g^2 (a+b \arcsin (c x))^2}{32 b c^3}+\frac {1}{4} f^2 x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {3}{8} f^2 x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {2 f g \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^2}-\frac {g^2 x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{16 c^2}+\frac {1}{6} g^2 x^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {1}{8} g^2 x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {3 f^2 (a+b \arcsin (c x))^2}{16 b c}+\frac {1}{16} b c^3 f^2 x^4+\frac {2}{25} b c^3 f g x^5+\frac {1}{36} b c^3 g^2 x^6-\frac {5}{16} b c f^2 x^2-\frac {4}{15} b c f g x^3+\frac {2 b f g x}{5 c}-\frac {7}{96} b c g^2 x^4+\frac {b g^2 x^2}{32 c}\right )}{\sqrt {1-c^2 x^2}}\) |
(d*Sqrt[d - c^2*d*x^2]*((2*b*f*g*x)/(5*c) - (5*b*c*f^2*x^2)/16 + (b*g^2*x^ 2)/(32*c) - (4*b*c*f*g*x^3)/15 + (b*c^3*f^2*x^4)/16 - (7*b*c*g^2*x^4)/96 + (2*b*c^3*f*g*x^5)/25 + (b*c^3*g^2*x^6)/36 + (3*f^2*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/8 - (g^2*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(16*c ^2) + (g^2*x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/8 + (f^2*x*(1 - c^2* x^2)^(3/2)*(a + b*ArcSin[c*x]))/4 + (g^2*x^3*(1 - c^2*x^2)^(3/2)*(a + b*Ar cSin[c*x]))/6 - (2*f*g*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(5*c^2) + (3*f^2*(a + b*ArcSin[c*x])^2)/(16*b*c) + (g^2*(a + b*ArcSin[c*x])^2)/(32*b *c^3)))/Sqrt[1 - c^2*x^2]
3.1.37.3.1 Defintions of rubi rules used
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_ ) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] & & EqQ[c^2*d + e, 0] && IGtQ[m, 0] && IntegerQ[p + 1/2] && GtQ[d, 0] && IGtQ [n, 0] && (m == 1 || p > 0 || (n == 1 && p > -1) || (m == 2 && p < -2))
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_ ) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[Simp[(d + e*x^2)^p/(1 - c^2*x^2)^ p] Int[(f + g*x)^m*(1 - c^2*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] /; FreeQ [{a, b, c, d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && Intege rQ[p - 1/2] && !GtQ[d, 0]
Result contains complex when optimal does not.
Time = 0.56 (sec) , antiderivative size = 1535, normalized size of antiderivative = 2.26
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1535\) |
| parts | \(\text {Expression too large to display}\) | \(1535\) |
a*(f^2*(1/4*x*(-c^2*d*x^2+d)^(3/2)+3/4*d*(1/2*x*(-c^2*d*x^2+d)^(1/2)+1/2*d /(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))))+g^2*(-1/6*x* (-c^2*d*x^2+d)^(5/2)/c^2/d+1/6/c^2*(1/4*x*(-c^2*d*x^2+d)^(3/2)+3/4*d*(1/2* x*(-c^2*d*x^2+d)^(1/2)+1/2*d/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d* x^2+d)^(1/2)))))-2/5*f*g/c^2/d*(-c^2*d*x^2+d)^(5/2))+b*(-1/32*(-d*(c^2*x^2 -1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^3/(c^2*x^2-1)*arcsin(c*x)^2*(6*c^2*f^2+g^2 )*d-1/2304*(-d*(c^2*x^2-1))^(1/2)*(-32*I*(-c^2*x^2+1)^(1/2)*c^6*x^6+32*c^7 *x^7+48*I*(-c^2*x^2+1)^(1/2)*x^4*c^4-64*c^5*x^5-18*I*(-c^2*x^2+1)^(1/2)*x^ 2*c^2+38*c^3*x^3+I*(-c^2*x^2+1)^(1/2)-6*c*x)*g^2*(I+6*arcsin(c*x))*d/c^3/( c^2*x^2-1)-1/400*(-d*(c^2*x^2-1))^(1/2)*(16*c^6*x^6-28*c^4*x^4-16*I*(-c^2* x^2+1)^(1/2)*x^5*c^5+13*c^2*x^2+20*I*(-c^2*x^2+1)^(1/2)*x^3*c^3-5*I*(-c^2* x^2+1)^(1/2)*x*c-1)*f*g*(I+5*arcsin(c*x))*d/c^2/(c^2*x^2-1)-1/512*(-d*(c^2 *x^2-1))^(1/2)*(-8*I*(-c^2*x^2+1)^(1/2)*x^4*c^4+8*c^5*x^5+8*I*(-c^2*x^2+1) ^(1/2)*x^2*c^2-12*c^3*x^3-I*(-c^2*x^2+1)^(1/2)+4*c*x)*(8*arcsin(c*x)*c^2*f ^2+2*I*c^2*f^2-4*arcsin(c*x)*g^2-I*g^2)*d/c^3/(c^2*x^2-1)-1/8*(-d*(c^2*x^2 -1))^(1/2)*(c^2*x^2-I*(-c^2*x^2+1)^(1/2)*x*c-1)*f*g*(arcsin(c*x)+I)*d/c^2/ (c^2*x^2-1)-1/8*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1 )*f*g*(arcsin(c*x)-I)*d/c^2/(c^2*x^2-1)+1/256*(-d*(c^2*x^2-1))^(1/2)*(2*I* (-c^2*x^2+1)^(1/2)*x^2*c^2+2*c^3*x^3-I*(-c^2*x^2+1)^(1/2)-2*c*x)*(-16*I*c^ 2*f^2+32*arcsin(c*x)*c^2*f^2-I*g^2+2*arcsin(c*x)*g^2)*d/c^3/(c^2*x^2-1)...
\[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )} \,d x } \]
integral(-(a*c^2*d*g^2*x^4 + 2*a*c^2*d*f*g*x^3 - 2*a*d*f*g*x - a*d*f^2 + ( a*c^2*d*f^2 - a*d*g^2)*x^2 + (b*c^2*d*g^2*x^4 + 2*b*c^2*d*f*g*x^3 - 2*b*d* f*g*x - b*d*f^2 + (b*c^2*d*f^2 - b*d*g^2)*x^2)*arcsin(c*x))*sqrt(-c^2*d*x^ 2 + d), x)
Timed out. \[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x)) \, dx=\text {Timed out} \]
\[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )} \,d x } \]
1/8*(2*(-c^2*d*x^2 + d)^(3/2)*x + 3*sqrt(-c^2*d*x^2 + d)*d*x + 3*d^(3/2)*a rcsin(c*x)/c)*a*f^2 + 1/48*a*g^2*(2*(-c^2*d*x^2 + d)^(3/2)*x/c^2 - 8*(-c^2 *d*x^2 + d)^(5/2)*x/(c^2*d) + 3*sqrt(-c^2*d*x^2 + d)*d*x/c^2 + 3*d^(3/2)*a rcsin(c*x)/c^3) - 2/5*(-c^2*d*x^2 + d)^(5/2)*a*f*g/(c^2*d) + sqrt(d)*integ rate(-(b*c^2*d*g^2*x^4 + 2*b*c^2*d*f*g*x^3 - 2*b*d*f*g*x - b*d*f^2 + (b*c^ 2*d*f^2 - b*d*g^2)*x^2)*sqrt(c*x + 1)*sqrt(-c*x + 1)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)), x)
Exception generated. \[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x)) \, dx=\text {Exception raised: RuntimeError} \]
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve cteur & l) Error: Bad Argument Value
Timed out. \[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x)) \, dx=\int {\left (f+g\,x\right )}^2\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2} \,d x \]